Question

If you set up the seventh harmonic on a string, (a) how many nodes are present, and (b) is there a node, antinode, or some intermediate state at the midpoint? If you next set up the sixth harmonic, (c) is its resonant wavelength longer or shorter than that for the seventh harmonic, and (d) is the resonant frequency higher or lower?

Short answer

1. For the seventh harmonic, there are ‘eight’ nodes.
2. There willbe an‘antinode’ at the midpoint.
3. For the sixth harmonic, there are ‘seven’ nodes.
4. There will be a ‘node’ at the midpoint.

Step-by-step solution

Given

The set up has the seventh harmonic on a string

Determining the concept

Determine the number of nodes by knowing the number of harmonics of the standing wave.

(a) Determining the number of nodes for the seventh harmonic

Determine the number of nodes for the sixth and seventh harmonic as, for the nth harmonic there are (n+1) nodes. Also, for an even number of harmonics, there is ‘node’ at the midpoint and for the odd number of harmonic; there is ‘antinode’ at the midpoint.

for the n^th harmonic, there are (n+1) number of nodes. So, for the seventh harmonic there will be eight nodes.

Hence, for the seventh harmonic, there

(b) Determining is there a node, antinode or some intermediate state at the midpoint of the seventh harmonic

For an even number ofharmonics, there is ‘node’ at the midpoint and for the odd number of harmonic, there is ‘antinode’ at the midpoint.

So, for the seventh (odd) harmonic, there is an antinode at the midpoint.

Hence, there will be an ‘antinode’ at the midpoint.

(c) Determining the number of nodes for the sixth harmonic

Using similar logic of part a), for the sixth harmonic, there are ‘Seven’ nodes.

Hence, for the sixth harmonic, there are ‘seven’ nodes

(d) Determining is there a node, antinode or some intermediate state at the midpoint of the sixth harmonic

Applying the same logic of part b), thereis a‘node’ at the midpoint.

Hence, there willbe a‘node’ at the midpoint.

Therefore, determine the number of nodes by knowing the number of harmonics of the standing wave.